An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates

  • Ivan Bjerre Damgård
  • Gudmund Skovbjerg Frandsen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2751)


We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to an extends the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability, namely 256/331776 t for t iterations of the test in the worst case. We give bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2− 143 for k=500, t=2. Compared to earlier similar results for the Miller-Rabin test, the results indicates that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ivan Bjerre Damgård
    • 1
  • Gudmund Skovbjerg Frandsen
    • 1
  1. 1.BRICS, Department of Computer ScienceUniversity of Aarhus 

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